Similarly to [1]}, our strategy to tackle Q1, Q2 and Q3 is essentially geometric and physically consistent with the assumptions made to derive the system (REF ). For a certain fixed time \(t\)  we firstly construct an auxiliary function that generalises the one introduced in [2]} and use the non-locality of the convolution to extend to a straight line/circle local properties of the solutions. Then we use one of the Hamiltonians to extend to any other time the result proved for the fixed \(t\) . Our approach can be interpreted as follows, from two different viewpoints:
